Schubert Classes Research Articles

Combinatorics of semi-toric degenerations of Schubert varieties in type C

An approach to Schubert calculus is to realize Schubert classes as concrete combinatorial objects such as Schubert polynomials. Using the polytope ring of the Gelfand–Tsetlin polytopes, Kiritchenko–Smirnov–Timorin realized each Schubert class as a sum of reduced Kogan faces. The first named author introduced a generalization of reduced Kogan faces to symplectic Gelfand–Tsetlin polytopes using a semi-toric degeneration of a Schubert variety, and extended the result of Kiritchenko–Smirnov–Timorin to type C case. In this paper, we introduce a combinatorial model to this type C generalization using a kind of pipe dream with self-crossings. As an application, we prove that the type C generalization can be constructed by skew mitosis operators.

Structure constants in equivariant oriented cohomology of flag varieties

We introduce generalized Demazure operators for the equivariant oriented cohomology of the flag variety, which have specializations to various Demazure operators and Demazure–Lusztig operators in both equivariant cohomology and equivariant K-theory. In the context of the geometric basis of the equivariant oriented cohomology given by certain Bott–Samelson classes, we use these operators to obtain formulas for the structure constants arising in different bases. Specializing to divided difference operators and Demazure operators in singular cohomology and K-theory, we recover the formulas for structure constants of Schubert classes obtained in Goldin and Knutson (Pure Appl Math Q 17(4):1345–1385, 2021). Two specific specializations result in formulas for the the structure constants for cohomological and K-theoretic stable bases as well; as a corollary we reproduce a formula for the structure constants of the Segre–Schwartz–MacPherson basis previously obtained by Su (Math Zeitschrift 298:193–213, 2021). Our methods involve the study of the formal affine Demazure algebra, providing a purely algebraic proof of these results.

Quantum K-theory of incidence varieties

We prove a conjecture of Buch and Mihalcea in the case of the incidence variety X=Fl(1,n-1;n)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$X=\ extrm{Fl}\\hspace{0.55542pt}(1,n-1;n)$$\\end{document} and determine the structure of its (T-equivariant) quantum K-theory ring. Our results are an interplay between geometry and combinatorics. The geometric side concerns Gromov–Witten varieties of 3-pointed genus 0 stable maps to X with markings sent to Schubert varieties, while on the combinatorial side are formulas for the (equivariant) quantum K-theory ring of X. We prove that the Gromov–Witten variety is rationally connected when one of the defining Schubert varieties is a divisor and another is a point. This implies that the (equivariant) K-theoretic Gromov–Witten invariants defined by two Schubert classes and a Schubert divisor class can be computed in the ordinary (equivariant) K-theory ring of X. We derive a positive Chevalley formula for the equivariant quantum K-theory ring of X and a positive closed formula for Littlewood–Richardson coefficients in the non-equivariant quantum K-theory ring of X. The Littlewood–Richardson rule in turn implies that non-empty Gromov–Witten varieties given by Schubert varieties in general position have arithmetic genus 0.

Closed 𝑘-Schur Katalan functions as 𝐾-homology Schubert representatives of the affine Grassmannian

Recently, Blasiak–Morse–Seelinger introduced symmetric func- tions called Katalan functions, and proved that the K K -theoretic k k -Schur functions due to Lam–Schilling–Shimozono form a subfamily of the Katalan functions. They conjectured that another subfamily of Katalan functions called closed k k -Schur Katalan functions is identified with the Schubert structure sheaves in the K K -homology of the affine Grassmannian. Our main result is a proof of this conjecture. We also study a K K -theoretic Peterson isomorphism that Ikeda, Iwao, and Maeno constructed, in a nongeometric manner, based on the unipotent solution of the relativistic Toda lattice of Ruijsenaars. We prove that the map sends a Schubert class of the quantum K K -theory ring of the flag variety to a closed K K - k k -Schur Katalan function up to an explicit factor related to a translation element with respect to an antidominant coroot. In fact, we prove this map coincides with a map whose existence was conjectured by Lam, Li, Mihalcea, Shimozono, and proved by Kato, and more recently by Chow and Leung.

Grothendieck lines in 3d N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 2 SQCD and the quantum K-theory of the Grassmannian

We revisit the 3d GLSM computation of the equivariant quantum K-theory ring of the complex Grassmannian from the perspective of line defects. The 3d GLSM onto X = Gr(Nc, nf) is a circle compactification of the 3d N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 2 supersymmetric gauge theory with gauge group UNck,k+lNc\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \ extrm{U}{\\left({N}_c\\right)}_{k,k+l{N}_c} $$\\end{document} and nf fundamental chiral multiplets, for any choice of the Chern-Simons levels (k, l) in the ‘geometric window’. For k=Nc−nf2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ k={N}_c-\\frac{n_f}{2} $$\\end{document} and l = −1, the twisted chiral ring generated by the half-BPS lines wrapping the circle has been previously identified with the quantum K-theory ring QKT(X). We identify new half-BPS line defects in the UV gauge theory, dubbed Grothendieck lines, which flow to the structure sheaves of the (equivariant) Schubert varieties of X. They are defined by coupling N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 2 supersymmetric gauged quantum mechanics of quiver type to the 3d GLSM. We explicitly show that the 1d Witten index of the defect worldline reproduces the Chern characters for the Schubert classes, which are written in terms of double Grothendieck polynomials. This gives us a physical realisation of the Schubert-class basis for QKT(X). We then use 3d A-model techniques to explicitly compute QKT(X) as well as other K-theoretic enumerative invariants such as the topological metric. We also consider the 2d/0d limit of our 3d/1d construction, which gives us local defects in the 2d GLSM, the Schubert defects, that realise equivariant quantum cohomology classes.

Equivariant K-Theory and Tangent Spaces to Schubert Varieties

Tangent spaces to Schubert varieties of type A were characterized by Lakshmibai and Seshadri [LS84]. This result was extended to the other classical types by Lakshmibai [Lak95, Lak00b], and [Lak00a]. We give a uniform characterization of tangent spaces to Schubert varieties in cominuscule G/P. Our results extend beyond cominuscule G/P; they describe the tangent space to any Schubert variety in G/B at a point xB, where x is a cominuscule Weyl group element in the sense of Peterson. Our results also give partial information about the tangent space to any Schubert variety at any point. Our method is to describe the tangent spaces of Kazhdan–Lusztig varieties, and then to recover results for Schubert varieties. Our proof uses a relationship between weights of the tangent space of a variety with torus action and factors of the class of the variety in torus equivariant K-theory. The proof relies on a formula for Schubert classes in equivariant K-theory due to Graham [Gra02] and Willems [Wil06] and on a theorem on subword complexes due to Knutson and Miller [KM04, KM05].

A Generalization of the Murnaghan–Nakayama Rule for K-k-Schur and k-Schur Functions

Abstract The $K$-$k$-Schur functions and $k$-Schur functions appeared in the study of $K$-theoretic and affine Schubert Calculus as polynomial representatives of Schubert classes. In this paper, we introduce a new family of symmetric functions $\mathcal {F}_{\lambda }^{(k)}$, that generalizes the constructions via the Pieri rule of $K$-$k$-Schur functions and $ k$-Schur functions. Then we obtain the Murnaghan–Nakayama rule for the generalized functions. The rule is described explicitly in the cases of $K$-$k$-Schur functions and $k$-Schur functions, with concrete descriptions and algorithms for coefficients. Our work recovers the result of Bandlow, Schilling, and Zabrocki for $k$-Schur functions, and explains it as a degeneration of the rule for $K$-$k$-Schur functions. In particular, many other special cases and connections promise to be detailed in the future.

A representation-theoretic interpretation of positroid classes

A positroid is the matroid of a real matrix with nonnegative maximal minors, a positroid variety is the closure of the locus of points in a complex Grassmannian whose matroid is a fixed positroid, and a positroid class is the cohomology class Poincaré dual to a positroid variety. We define a family of representations of general linear groups whose characters are symmetric polynomials representing positroid classes. These representations are certain diagram Schur modules in the sense of James and Peel. This gives a new algebraic interpretation of the Schubert structure constants for the product of a Schubert polynomial and Schur polynomial, and of the 3-point Gromov-Witten invariants for Grassmannians, proving a conjecture of Postnikov. As a byproduct, we obtain an effective algorithm for decomposing positroid classes into Schubert classes.

Inequalities of Chern classes on nonsingular projective $n$-folds with ample canonical or anti-canonical line bundles

Let $X$ be a nonsingular projective $n$-fold $(n \geq 2)$ which is either Fano or of general type with ample canonical bundle $K_X$ over an algebraic closed field $\kappa$ of any characteristic. We produce a new method to give a bunch of inequalities in terms of all the Chern classes $c_1, c_2, \dotsc , c_n$ by pulling back Schubert classes in the Chow group of Grassmannian under the Gauss map. Moreover, we show that if the characteristic of $\kappa$ is $0$, then the Chern ratios $\left( \dfrac{c_{2,1^{n-2}}}{c_{1^n}} , \dfrac{c_{2,2,1^{n-4}}}{c_{1^n}} , \dotsc , \frac{c_n}{c_{1^n}} \right)$ are contained in a convex polyhedron depending on the dimension of $X$ only. So we give an affirmative answer to a generalized open question, that whether the region described by the Chern ratios is bounded, posted by Hunt [Hun] to all dimensions. As a corollary, we can get that there exist constants $d_1$, $d_2$, $d_3$ and $d_4$ depending only on $n$ such that $d_1 K^n_X \leq \chi_\mathrm{top} (X) \leq d_2 K^n_X$ and $d_3 K^n_X \leq \chi (X, \mathcal{O}_X) \leq d_4 K^n_X$ . If the characteristic of $\kappa$ is positive, $K_X$ (or $-K_X$) is ample and $\mathcal{O}_X (K_X) (\mathcal{O}_X(-K_X) \textrm{, respectively})$ is globally generated, then the same results hold.

Pipe dreams for Schubert polynomials of the classical groups

Schubert polynomials for the classical groups were defined by S.Billey and M.Haiman in 1995; they are polynomial representatives of Schubert classes in a full flag variety over a classical group. We provide a combinatorial description for these polynomials, as well as their double versions, by introducing analogues of pipe dreams, or RC-graphs, for Weyl groups of classical types.

A Positive Formula for Type A Peterson Schubert Calculus

The Peterson variety is a special case of a nilpotent Hessenberg variety, a class of subvarieties of G/B that have appeared in the study of quantum cohomology, representation theory and combinatorics. In type A, the Peterson variety Y is a subvariety of $$Fl(n;{\mathbb {C}})$$ , the set of complete flags in $${\mathbb {C}}^n$$ , and comes equipped with an action by a one-dimensional torus subgroup S of a standard torus T that acts on $$Fl(n;{\mathbb {C}})$$ . Using the Peterson Schubert basis introduced in Harada and Tymoczko (Proc Lond Math Soc 103(1):40–72, 2011) and obtained by restricting a specific set of Schubert classes from $$H_T^*(Fl(n; {\mathbb {C}}))$$ to $$H_S^*(Y)$$ , we describe the product structure of the equivariant cohomology $$H_{S}^*(Y)$$ . In particular, we show that the product is manifestly positive in an appropriate sense by providing an explicit, positive, combinatorial formula for its structure constants. A key step in our proof requires a new combinatorial identity of binomial coefficients that generalizes Vandermonde’s identity, and merits independent interest.

Minimum Quantum Degrees for Isotropic Grassmannians in Types B and C

We give a formula in terms of Young diagrams to calculate the minimum positive integer d such that $$q^d$$ appears in the quantum product of two Schubert classes for the submaximal isotropic Grassmannians in types B and C. We do this by studying curve neighborhoods. We compute curve neighborhoods in several combinatorial models including k-strict partitions and a set of partitions where their inclusion is compatible with the Bruhat order.

An Equivariant Quantum Pieri Rule for the Grassmannian on Cylindric Shapes

The quantum cohomology ring of the Grassmannian is determined by the quantum Pieri rule for multiplying by Schubert classes indexed by row or column-shaped partitions. We provide a direct equivariant generalization of Postnikov's quantum Pieri rule for the Grassmannian in terms of cylindric shapes, complementing related work of Gorbounov and Korff in quantum integrable systems. The equivariant terms in our Graham-positive rule simply encode the positions of all possible addable boxes within one cylindric skew diagram. As such, unlike the earlier equivariant quantum Pieri rule of Huang and Li and known equivariant quantum Littlewood-Richardson rules, our formula does not require any calculations in a different Grassmannian or two-step flag variety.

Schubert calculus from polyhedral parametrizations of Demazure crystals

One approach to Schubert calculus is to realize Schubert classes as concrete combinatorial objects such as Schubert polynomials. Through an identification of the cohomology ring of the type A full flag variety with the polytope ring of the Gelfand–Tsetlin polytopes, Kiritchenko–Smirnov–Timorin realized each Schubert class as a sum of reduced (dual) Kogan faces. In this paper, we explicitly describe string parametrizations of opposite Demazure crystals, which give a natural generalization of reduced dual Kogan faces. We also relate reduced Kogan faces with Demazure crystals using the theory of mitosis operators, and apply these observations to develop the theory of Schubert calculus on symplectic Gelfand–Tsetlin polytopes.

The Permutahedral Variety, Mixed Eulerian Numbers, and Principal Specializations of Schubert Polynomials

AbstractWe compute the expansion of the cohomology class of the permutahedral variety in the basis of Schubert classes. The resulting structure constants $a_w$ are expressed as a sum of normalized mixed Eulerian numbers indexed naturally by reduced words of $w$. The description implies that the $a_w$ are positive for all permutations $w\in S_n$ of length $n-1$, thereby answering a question of Harada, Horiguchi, Masuda, and Park. We use the same expression to establish the invariance of $a_w$ under taking inverses and conjugation by the longest word and subsequently establish an intriguing cyclic sum rule for the numbers. We then move toward a deeper combinatorial understanding for the $a_w$ by exploiting in addition the relation to Postnikov’s divided symmetrization. Finally, we are able to give a combinatorial interpretation for $a_w$ when $w$ is vexillary, in terms of certain tableau descents. It is based in part on a relation between the $a_w$ and principal specializations of Schubert polynomials. Along the way, we prove results and raise questions of independent interest about the combinatorics of permutations, Schubert polynomials, and related objects. We also sketch how to extend our approach to other Lie types, highlighting an identity of Klyachko in particular.

Neutral-fermionic presentation of the K-theoretic Q-function

We show a new neutral-fermionic presentation of Ikeda-Naruse's $K$-theoretic $Q$-functions, which represent a Schubert class in the $K$-theory of coherent sheaves on the Lagrangian Grassmannian. Our presentation provides a simple description and yields straightforward proof of two types of Pfaffian formulas for them. We present a dual space of $G\Gamma$, the vector space generated by all $K$-theoretic $Q$-functions, by constructing a non-degenerate bilinear form that is compatible with the neutral fermionic presentation. We give a new family of dual $K$-theoretic $Q$-functions, their neutral-fermionic presentations, and Pfaffian formulas.

Schubert Class and Cyclotomic NilHecke Algebras

Let [Formula: see text] and [Formula: see text] be positive integers such that [Formula: see text], and let [Formula: see text] be the Grassmannian which consists of the set of [Formula: see text]-dimensional subspaces of [Formula: see text]. There is a [Formula: see text]-graded algebra isomorphism between the cohomology [Formula: see text] of [Formula: see text] and a natural [Formula: see text]-form [Formula: see text] of the [Formula: see text]-graded basic algebra of the type [Formula: see text] cyclotomic nilHecke algebra [Formula: see text]. We show that the isomorphism can be chosen such that the image of each (geometrically defined) Schubert class [Formula: see text] coincides with the basis element [Formula: see text] constructed by Hu and Liang by purely algebraic method, where [Formula: see text] with [Formula: see text] for each [Formula: see text], and [Formula: see text] is the [Formula: see text]-multipartition of [Formula: see text] associated to [Formula: see text]. A similar correspondence between the Schubert class basis of the cohomology of the Grassmannian [Formula: see text] and the [Formula: see text]'s basis ([Formula: see text] is an [Formula: see text]-multipartition of [Formula: see text] with each component being either [Formula: see text] or empty) of the natural [Formula: see text]-form [Formula: see text] of the [Formula: see text]-graded basic algebra of [Formula: see text] is also obtained. As an application, we obtain a second version of the Giambelli formula for Schubert classes.

Curve neighborhoods and minimal degrees in quantum products

Let G be a connected, simply connected, simple, complex, linear algebraic group. Let P be an arbitrary parabolic subgroup of G. Let be the G-homogeneous projective space attached to this situation. We consider the (small) quantum cohomology ring attached to X. Postnikov proved that any quantum product of two Schubert classes admits a unique minimal degree. In particular, this result implies that there exists a unique degree d which is minimal with the property that qd occurs with nonzero coefficient in the quantum product of two point classes. This unique minimal degree in is denoted by dX . We use the technique of curve neighborhoods by Buch-Mihalcea to compute dX explicitly in terms of Kostant’s cascade of orthogonal roots. Moreover, we prove orthogonality relations in greedy decompositions of minimal degrees. Such orthogonality relations can be seen as a generalization of the computation of dX from one minimal degree to every minimal degree, and were successfully applied to prove quasi-homogeneity of the moduli space of stable maps.

An equivariant basis for the cohomology of Springer fibers

Springer fibers are subvarieties of the flag variety that play an important role in combinatorics and geometric representation theory. In this paper, we analyze the equivariant cohomology of Springer fibers for G L n ( C ) GL_n(\mathbb {C}) using results of Kumar and Procesi that describe this equivariant cohomology as a quotient ring. We define a basis for the equivariant cohomology of a Springer fiber, generalizing a monomial basis of the ordinary cohomology defined by De Concini and Procesi and studied by Garsia and Procesi. Our construction yields a combinatorial framework with which to study the equivariant and ordinary cohomology rings of Springer fibers. As an application, we identify an explicit collection of (equivariant) Schubert classes whose images in the (equivariant) cohomology ring of a given Springer fiber form a basis.

A critical point analysis of Landau–Ginzburg potentials with bulk in Gelfand–Cetlin systems

Using the bulk deformation of Floer cohomology by Schubert classes and non-Archimedean analysis of Fukaya–Oh–Ohta–Ono’s bulk-deformed potential function, we prove that every complete flag manifold Fl(n) (n≥3) with a monotone Kirillov–Kostant–Souriau (KKS) symplectic form carries a continuum of nondisplaceable Lagrangian tori which degenerates to a nontorus fiber in the Hausdorff limit. In particular, the Lagrangian S3-fiber in Fl(3) is nondisplaceable, answering a question raised by Nohara and Ueda who computed its Floer cohomology to be vanishing.